Structural Optimization — The Interaction between Form and Mechanics

Abstract

Usually mechanical laws are applied to determine the structural response, for example deflections and stress state, while loads, boundary conditions, and geometry of the structure, i.e. the topology and the shape, are given. However, the mechanical principles can also be used to determine topology and shape of a structure for a prescribed structural response. This inverse method is called structural optimization. Since structural optimization deals in general with nonlinear and implicit functionals, only numerical methods have a chance to solve application-orientated problems in engineering design. Structural optimization can be distinguished into material, shape, and topology optimization depending on what is varied in the optimization process. The most challenging task is to determine the basic geometrical layout by topology optimization. In particular, recently the so-called material topology optimization of continuous structures has gained substantial interest both by mathematicians as well as engineers. The present contribution tries to consolidate these developments from an engineering point of view. In order to overcome problems of conventional numerical modeling techniques, material topology optimization is extended to geometrically adaptive methods. The adaptive discretization of the geometrical model in topology optimization turns out to be an appropriate way to obtain reliable results and simultaneously to reduce the numerical effort. This is verified by topology optimization problems with stress constraints and considering elastoplastic material behavior. The optimization based on either a linear or a nonlinear structural response leads to completely different results and shows the relevance of an appropriate mechanical model in the optimization process.